The Numerical Recipes Forum http://numerical.recipes/forum/ is closed, so I will record a tiny typo here for the benefit of others who may wonder about this. (This typo is not in the software, but only in the text, and is currently in the online version of the textbook at http://numerical.recipes/book):
In Eq. 9.4.6 (describing the convergence rate of the Newton-Raphson algorithm), the "-" sign on the RHS should be removed. This can be easily verified by substituting 9.4.3 into 9.4.5 and expanding for small epsilon_i to keep the lowest order non-zero term. One can also see from Fig. 9.4.1 that if f' and f'' are positive then a Newton method converges from the right (positive errors).
A possible source of confusion in looking at other references is that some people define an error eps_j in an estimate x_j as eps_j = x - x_j, where x is the exact answer, but the more traditional definition (as used in most statistics references) is eps_j = x_j - x. Numerical Recipes uses the more traditional definition in Eq. 9.4.3, so the "-" sign in 9.4.6 is wrong.